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 advanced_tools:group_theory:representation_theory:adjoint_representation [2020/11/29 16:59]edi [Concrete] advanced_tools:group_theory:representation_theory:adjoint_representation [2020/12/26 22:49]edi [Concrete] Both sides previous revision Previous revision 2020/12/26 22:49 edi [Concrete] 2020/11/29 16:59 edi [Concrete] 2020/11/29 16:24 edi [Intuitive] 2018/04/15 16:38 aresmarrero [Concrete] 2018/04/15 16:38 aresmarrero [Why is it interesting?] 2018/04/15 16:35 aresmarrero [Concrete] 2018/04/15 16:34 aresmarrero created 2020/12/26 22:49 edi [Concrete] 2020/11/29 16:59 edi [Concrete] 2020/11/29 16:24 edi [Intuitive] 2018/04/15 16:38 aresmarrero [Concrete] 2018/04/15 16:38 aresmarrero [Why is it interesting?] 2018/04/15 16:35 aresmarrero [Concrete] 2018/04/15 16:34 aresmarrero created Line 14: Line 14: **Example** **Example** - The diagram below shows the defining representation of $SU(2)$ in its upper branch. To construct the adjoint representation,​ we use the Lie algebra as the representation space, as shown in the lower branch (red arrows). The group elements act on this space like $L'​=ULU^{-1}$ and the Lie-algebra elements like $L'​=[J,​L]$. It is possible to rewrite ​this representation such that it acts on 3-dimensional vectors (as opposed to 3x3 matrices) by regular matrix-vector multiplication. + The diagram below shows the defining representation of $SU(2)$ in its upper branch. To construct the adjoint representation,​ we use the Lie algebra as the representation space, as shown in the lower branch (red arrows). The group elements act on this space like $L'​=ULU^{-1}$ and the Lie-algebra elements ​act like $L'​=[J,​L]$. It is possible to rewrite ​the adjoint ​representation such that it acts on 3-dimensional vectors (as opposed to 2x2 matrices) by regular matrix-vector multiplication. [{{ :​advanced_tools:​group_theory:​representation_theory:​su2_adjoint.jpg?​nolink }}] [{{ :​advanced_tools:​group_theory:​representation_theory:​su2_adjoint.jpg?​nolink }}]